## (a) Show that for a projectile the angle between the velocity

[question] Question. (a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is given by $\theta(t)=\tan ^{-1}\left(\frac{v_{0 y}-\mathrm{g} t}{v_{0 x}}\right)$ (b) Show that the projection angle $\theta_{0}$ for a projectile launched from the origin is given by $\theta_{0}=\tan ^{-1}\left(\frac{4 h_{m}}{R}\right)$ Where the symbols have their usual meaning. [/question] [solution] solution: (a) Let $v_{0 x}$ and $v_{0 y}$ respectively be the initial com...

## Can you associate vectors with (a) the length of a wire bent into a loop,

[question] Question. Can you associate vectors with (a) the length of a wire bent into a loop, (b) a plane area, (c) a sphere? Explain. [/question] [solution] solution: Answer: No; Yes; No (a) One cannot associate a vector with the length of a wire bent into a loop. (b) One can associate an area vector with a plane area. The direction of this vector is normal, inward or outward to the plane area. (c) One cannot associate a vector with the volume of a sphere. However, an area vector can be associ...

## A vector has magnitude and direction.

[question] Question. A vector has magnitude and direction. Does it have a location in space? Can it vary with time? Will two equal vectors a and b at different locations in space necessarily have identical physical effects? Give examples in support of your answer. [/question] [solution] solution: Answer: No; Yes; No Generally speaking, a vector has no definite locations in space. This is because a vector remains invariant when displaced in such a way that its magnitude and direction remain the s...

## An aircraft is flying at a height of 3400 m above the ground.

[question] Question. An aircraft is flying at a height of $3400 \mathrm{~m}$ above the ground. If the angle subtended at a ground observation point by the aircraft positions $10.0 \mathrm{~s}$ apart is $30^{\circ}$, what is the speed of the aircraft? [/question] [solution] solution: The positions of the observer and the aircraft are shown in the given figure. Height of the aircraft from ground, OR = 3400 m Angle subtended between the positions, $\angle \mathrm{POQ}=30^{\circ}$ Time = 10 s In $\t...

## Read each statement below carefully and state,

[question] Question. Read each statement below carefully and state, with reasons and examples, if it is true or false: A scalar quantity is one that (a) is conserved in a process (b) can never take negative values (c) must be dimensionless (d) does not vary from one point to another in space (e) has the same value for observers with different orientations of axes [/question] [solution] solution: (a) False Despite being a scalar quantity, energy is not conserved in inelastic collisions. (b) False...

## For any arbitrary motion in space, which of the following relations are true:

[question] Question. For any arbitrary motion in space, which of the following relations are true: (a) $\mathbf{v}_{\text {aventes }}=\left(\frac{1}{2}\right)\left(\mathbf{v}\left(t_{1}\right)+\mathbf{v}\left(t_{2}\right)\right)$ (b) $\mathbf{v}_{\text {avenee }}=\frac{\left[\mathbf{r}\left(t_{2}\right)-\mathbf{r}\left(t_{1}\right)\right]}{\left(t_{2}-t_{1}\right)}$ (c) $\mathbf{v}(t)=\mathbf{v}(0)+\mathbf{a} t$ (d) $\mathbf{r}(t)=\mathbf{r}(0)+\mathbf{v}(0) t+\left(\frac{1}{2}\right) \mathbf{a}...

## The position of a particle is given by

[question] Question. The position of a particle is given by $\mathbf{r}=3.0 t \hat{\mathbf{i}}-2.0 t^{2} \hat{\mathbf{j}}+4.0 \hat{\mathbf{k}} \mathrm{m}$ Where $t$ is in seconds and the coefficients have the proper units for $r$ to be in metres. (a) Find the $\mathbf{v}$ and $\mathbf{a}$ of the particle? (b) What is the magnitude and direction of velocity of the particle at $t=2.0 \mathrm{~s}$ ? [/question] [solution] solution: (a) $\vec{v}(t)=(3.0 \hat{\mathbf{i}}-4.0 t \hat{\mathbf{j}}) ; \ve...

## Read each statement below carefully and state,

[question] Question. Read each statement below carefully and state, with reasons, if it is true or false: (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point (c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector [/question] [solution] solution: (a) False The ...

## An aircraft executes a horizontal loop of radius 1.00 km with

[question] Question. An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity. [/question] [solution] solution: Radius of the loop, $r=1 \mathrm{~km}=1000 \mathrm{~m}$ Speed of the aircraft, $v=900 \mathrm{~km} / \mathrm{h}=900 \times \frac{5}{18}=250 \mathrm{~m} / \mathrm{s}$ Centripetal acceleration, $a_{\mathrm{c}}=\frac{v^{2}}{r}$ $=\frac{(250)^{2}}{1000}=62.5 \mathrm{~m} / \mathrm{s}^{...

## A stone tied to the end of a string 80 cm long is whirled in a

[question] Question. A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone? [/question] [solution] solution: Length of the string, $I=80 \mathrm{~cm}=0.8 \mathrm{~m}$ Number of revolutions $=14$ Frequency, $v=\frac{\text { Number of revolutions }}{\text { Time taken }}=\frac{14}{25} \mathrm{~Hz}$ Angular frequency, $\omega=2 \pi \mathrm{V}$...

## A cricketer can throw a ball to a maximum horizontal distance of 100 m.

[question] Question. A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball? [/question] [solution] solution: Maximum horizontal distance, R = 100 m The cricketer will only be able to throw the ball to the maximum horizontal distance when the angle of projection is $45^{\circ}$, i.e., $\theta=45^{\circ}$. The horizontal range for a projection velocity $v$, is given by the relation: $R=\frac{u^{2} \sin 2 \theta}...

## The ceiling of a long hall is 25 m high.

[question] Question. The ceiling of a long hall is $25 \mathrm{~m}$ high. What is the maximum horizontal distance that a ball thrown with a speed of $40 \mathrm{~m} \mathrm{~s}^{-1}$ can go without hitting the ceiling of the hall? [/question] [solution] solution: Speed of the ball, $u=40 \mathrm{~m} / \mathrm{s}$ Maximum height, $h=25 \mathrm{~m}$ In projectile motion, the maximum height reached by a body projected at an angle $\theta$, is given by the relation: $h=\frac{u^{2} \sin ^{2} \theta}{...

## Rain is falling vertically with a speed of

[question] Question. Rain is falling vertically with a speed of $30 \mathrm{~m} \mathrm{~s}^{-1}$. A woman rides a bicycle with a speed of $10 \mathrm{~m} \mathrm{~s}^{-1}$ in the north to south direction. What is the direction in which she should hold her umbrella? [/question] [solution] solution: The described situation is shown in the given figure. Here, $v_{c}=$ Velocity of the cyclist $v_{\mathrm{r}}=$ Velocity of falling rain In order to protect herself from the rain, the woman must hold h...

## Given $\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}=0$, which of the following statements are correct:

[question] Question. Given $\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}=0$, which of the following statements are correct: (a) $\mathbf{a}, \mathbf{b}, \mathbf{c}$, and $\mathrm{d}$ must each be a null vector, (b) The magnitude of $(\mathbf{a}+\mathbf{c})$ equals the magnitude of $(\mathbf{b}+\mathbf{d})$, (c) The magnitude of a can never be greater than the sum of the magnitudes of $\mathbf{b}, \mathbf{c}$, and $\mathbf{d}$, (d) $\mathbf{b}+\mathbf{c}$ must lie in the plane of $\mathbf{a}$ and ...

## Establish the following vector inequalities geometrically or otherwise:

[question] Question. Establish the following vector inequalities geometrically or otherwise: (a) $|\mathbf{a}+\mathbf{b}| \leq|\mathbf{a}|+|\mathbf{b}|$ (b) $|\mathbf{a}+\mathbf{b}| \geq\|\mathbf{a}|-| \mathbf{b}\|$ (c) $|\mathbf{a}-\mathbf{b}| \leq|\mathbf{a}|+|\mathbf{b}|$ (d) $|\mathbf{a}-\mathbf{b}| \geq|| \mathbf{a}|-| \mathbf{b}||$ When does the equality sign above apply? [/question] [solution] solution: (a) Let two vectors $\vec{a}$ and $\vec{b}$ be represented by the adjacent sides of a ...

## Pick out the only vector quantity in the following list: Temperature, pressure,

[question] Question. Pick out the only vector quantity in the following list: Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge. [/question] [solution] solution: Impulse Impulse is given by the product of force and time. Since force is a vector quantity, its product with time (a scalar quantity) gives a vector quantity. [/solution]...

## Pick out the two scalar quantities in the following list:

[question] Question. Pick out the two scalar quantities in the following list: force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity. [/question] [solution] solution: Work and current are scalar quantities. Work done is given by the dot product of force and displacement. Since the dot product of two quantities is always a scalar, work is a scalar physical quantity. Current is described only by its magnitude. Its direction is...

## Pick out the two scalar quantities in the following list:

[question] Question. Pick out the two scalar quantities in the following list: force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity. [/question] [solution] solution: calar: Volume, mass, speed, density, number of moles, angular frequency Vector: Acceleration, velocity, displacement, angular velocity A scalar quantity is specified by its magnitude only. It does not have any direction associated with it. Volume, mass, speed, ...

## State, for each of the following physical quantities,

[question] Question. State, for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity. [/question] [solution] solution: Scalar: Volume, mass, speed, density, number of moles, angular frequency Vector: Acceleration, velocity, displacement, angular velocity A scalar quantity is specified by its magnitude only. It does not have any direction associated with it...